Finite Group Databasebeta

← #28299102

Subgroups — #28299102

Order 13568.

224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20289809C211yesmaximal, characteristic
2#20289644C211yesmaximal, characteristic
3#20276249C211yesmaximal, characteristic
4C2×#13133276—21no—
5#16811468—21no—
6C2×#13133148—21no—
7C2×#13133137—21no—
8#16810525C2211yescharacteristic
9#16810378C411yescharacteristic, complemented
10C53×#384C411yescharacteristic, complemented
11C2×C22⋊#6157—41no—
12#13133276—22no—
13#13133276—41no—
14C22⋊#88324—41no—
15C22×D212D411yescharacteristic
16C22×#88322—21no—
17C2×C4⋊Dic53D411yescharacteristic
18C2×#11719921—41no—
19#13133148—42no—
20#13133137—42no—
21C2×C4⋊C4×C53—21no—
22C2×C4⋊C4×C53—41no—
23C2×C42×C53C2×C411yescharacteristic
24C23×D53—21no—
25C22×#6157—81no—
26C22×C4×C53—21no—
27C22×C4×C53—41no—
28C22×C4×C53C22⋊C411yescharacteristic
29C22×Dic53—21no—
30C2×D212—21no—
31C2×D212—41no—
32C2×D212—41no—
33C2×D212SD1611yescharacteristic
34C2×D212Q1611yescharacteristic
35C22⋊#6157—42no—
36C22⋊#6157—81no—
37C2×#88324—81no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—82no—
41C42×C53—41no—
42#11719921—82no—
43C4⋊Dic53—41no—
44C2×#88322—22no—
45C2×#88322—41no—
46C2×#88322—41no—
47C23×C53#9911yescharacteristic
48C22×D53—21no—
49C22×D53—21no—
50C22×D53—41no—
51C22×D53—41no—
52C22×D53—81no—
53C22×D53—81no—
54C2×#6157—82no—
55C2×#6157—161no—
56C2×#6157—161no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—82no—
63C2×C4×C53—81no—
64C2×C4×C53#10311yescharacteristic
65C2×C4×C53#10411yescharacteristic
66C2×Dic53—41no—
67C2×Dic53—81no—
68D212—41no—
69D212—41no—
70D212—81no—
71#88324—161no—
72#88322—42no—
73#7435—531nomaximal, Sylow
74C22×C53—21no—
75C22×C53—21no—
76C22×C53#32811yescharacteristic
77C22×C53#35211yescharacteristic
78C22×C53#35311yescharacteristic
79C2×D53—41no—
80C2×D53—41no—
81C2×D53—81no—
82C2×D53—81no—
83C2×D53—81no—
84C2×D53—81no—
85C2×D53—161no—
86#6157—162no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—42no—
90C4×C53—161no—
91Dic53—81no—
92#2094—531no—
93#1393—531no—
94#1318—531no—
95C2×C53—21no—
96C2×C53—21no—
97C2×C53#135211yescharacteristic
98C2×C53#140611yescharacteristic
99C2×C53#140911yescharacteristic
100D53—81no—
101D53—81no—
102C2×#106—1061no—
103C2×#102—1061no—
104C2×#99—1061no—
105#390—531no—
106#384#615711yescharacteristic, complemented
107#336—531no—
108#324—1061no—
109C53#743511yescharacteristic, complemented, Sylow
110C22×D4—531no—
111C22×C8—1061no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—531no—
115C2×C22⋊C4—2121no—
116C2×C22⋊C4—2121no—
117C2×C42C2×#615711yescharacteristic
118#106—2122no—
119#102—2122no—
120#99—1062no—
121#99—2121no—
122C22⋊C8—2121no—
123C24—1061no—
124C2×D4—531no—
125C2×D4—531no—
126C2×D4—1061no—
127C2×D4—2121no—
128C2×D4—2121no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—1061no—
132C22×C4—4241no—
133C22×C4C22⋊#615711yescharacteristic
134C2×C8—1062no—
135C2×C8—4241no—
136C2×C8—2121no—
137C2×C8—2121no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—82no—
141C4⋊C4—2121no—
142C22⋊C4—4242no—
143C22⋊C4—4241no—
144C22⋊C4—2122no—
145C42—41no—
146C23—1061no—
147C23—1061no—
148C23—4241no—
149C23—4241no—
150C23—2121no—
151C23—2121no—
152C23#1313327511yescharacteristic
153D4—4241no—
154D4—2121no—
155D4—2121no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4—8481no—
164C2×C4—8481no—
165C2×C4—4242no—
166C2×C4—4241no—
167C2×C4—2121no—
168C2×C4#1313328011yescharacteristic
169C2×C4#1313328211yescharacteristic
170C8—8481no—
171C8—2122no—
172C22—21no—
173C22—21no—
174C22—8481no—
175C22—4241no—
176C22—4241no—
177C22—4241no—
178C22—4241no—
179C22—2121no—
180C22—2121no—
181C22#1681147711yescharacteristic
182C22#1681152511yescharacteristic
183C22#1681152711yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—161no—
188C4—8482no—
189C4—4241no—
190C2—21no—
191C2—21no—
192C2—4241no—
193C2—4241no—
194C2#2028971211yescharacteristic
195C2#2028984211yescharacteristic
196C2#2028984811yescharacteristic
subgroupG/H
centerC22#16811527
commutatorC2×C42×C53C2×C4
FrattiniC2×C42C2×#6157
FittingC53×#384C4
socleC22×C53#353
prime radicalC22×D212D4
cosocle#16810525C22
hypercenter#384#6157
hypocenterC53#7435
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #384

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
9. #16810378C4semidirect
10. C53×#384C4semidirect
106. #384#6157semidirect
109. C53#7435semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202898092
#202896442
#202762492
#743553